Ultrarelativistic Electron States in a General Background Electromagnetic Field

Abstract

The feasibility of obtaining exact analytical results in the realm of QED in the presence of a background electromagnetic field is almost exclusively limited to a few tractable cases, where the Dirac equation in the corresponding background field can be solved analytically. This circumstance has restricted, in particular, the theoretical analysis of QED processes in intense laser fields to within the plane-wave approximation even at those high intensities, achievable experimentally only by tightly focusing the laser energy in space. Here, within the Wentzel-Kramers-Brillouin (WKB) or eikonal approximation, we construct analytically single-particle electron states in the presence of a background electromagnetic field of general space-time structure in the realistic assumption that the initial energy of the electron is the largest dynamical energy scale in the problem. The relatively compact expression of these states opens, in particular, the possibility of investigating analytically strong-field QED processes in the presence of spatially focused laser beams, which is of particular relevance in view of the upcoming experimental campaigns in this field. The predictions of QED have been confirmed with outstanding precision in numerous experiments. The impressive agreement between the theoretical and the experimental value of the electron (g−2)-factor is customarily quoted as a prominent example [1]. However, the experimental scrutiny of QED becomes much less thorough when processes are involved, occurring in the presence of a strong background electromagnetic field, i.e. of the order of F cr = m 2 c 3 / |e| = 1.3 × 10 16 V/cm = 4.4 × 10 13 G (here m and e < 0 are the electron mass and charge, respectively) [2]. The main reason is that these values largely exceed the field strengths available in laboratories. An important exception is represented by the electric field of highly-charged ions (charge number Z ∼ 1/α, with α = e 2 / c ≈ 1/137) at the typical QED length λ C = /mc = 3.9 × 10 −11 cm [3,4]. Indeed, numerous experiments on processes occurring in the presence of highly-charged ions [5][6][7][8] have already successfully confirmed the predictions of QED. Correspondingly, advanced analytical methods [9], have been developed to interpret accurate experimental data beyond the exactlysolvable Coulomb model of the ionic field.Modern high-power lasers represent an alternative source of intense electromagnetic fields structurally thoroughly different from atomic fields. Although the amplitude F 0 of the strongest laser pulse ever produced is about 10 −4 F cr [10], it can be boosted to an effective strength F * 0 ∼ F cr in the rest-frame of ultrarelativistic particles colliding with the laser beam [11]. This principle has been exploited at SLAC to perform the so-far unique experimental campaign on strong-laser field QED [12], employing a laser with photon energy 1 eV and amplitude F 0 = 2.7 × 10 10 V/cm, and an almost counter-propagating electron beam with energy of 45 GeV (F * 0 ≈ 0.3 F cr ). The relatively large pul...

Classical and quantum particle dynamics in univariate background fields

“…In order to gain analytical insights, confirm numerical results, and improve experimental predictions, we must go beyond the plane wave model [16][17][18][19][20][21][22][23]. This task is as challenging as it is open-ended.…”

Analytical solution for the Klein-Gordon equation and action function of the solution for the Dirac equation in counterpropagating laser waves

“…It has the advantage of clear structure and close similarity to the solution in a plane-wave case, and thus favors convenient application. With the solution of the Klein-Gordon equation, the solution of the corresponding Dirac equation in the same electromagnetic field can be derived by the method shown in paper [23]. It then can provide more exact reaction rates of multi-photon pair production process, Compton scattering process and so on in this non-plane-wave field.…”

“…This is referred to as the largest dynamical parameter condition in the paper similar to that in Ref. [23]. Besides this energy requirement, the geometry of the relative movements of the particle and the laser waves can become significant in special cases, which will be discussed at the end of this section.…”

“…which is just the equation determining the action function S(x) for Dirac equation [23,24]. In section 4 the newly obtained solution is justified by showing that solutions in a plane-wave field can be recovered naturally from it.…”

“…The recent breakthrough in solving the Dirac equation in propagating laser waves is worked out by A. Di Piazza in [23,24]. Applying the WKB method and looking for a solution of the form ψ(x) = exp[iS(x)/ ]ϕ(x), the action function S(x) is derived first by solving the classical electron dynamical equation in the background electromagnetic field and then the bi-spinor ϕ(x) is constructed via the method of characteristics.…”